Optimal. Leaf size=20 \[ \frac {3 \sqrt [3]{x^2-1}}{2 (x+1)^{2/3}} \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {651} \[ \frac {3 \sqrt [3]{x^2-1}}{2 (x+1)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin {align*} \int \frac {1}{(1+x)^{2/3} \left (-1+x^2\right )^{2/3}} \, dx &=\frac {3 \sqrt [3]{-1+x^2}}{2 (1+x)^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 23, normalized size = 1.15 \[ \frac {3 (x-1) \sqrt [3]{x+1}}{2 \left (x^2-1\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 14, normalized size = 0.70 \[ \frac {3 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}}}{2 \, {\left (x + 1\right )}^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 13, normalized size = 0.65 \[ \frac {3}{2} \, {\left (-\frac {2}{x + 1} + 1\right )}^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.90 \[ \frac {3 \left (x -1\right ) \left (x +1\right )^{\frac {1}{3}}}{2 \left (x^{2}-1\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{2} - 1\right )}^{\frac {2}{3}} {\left (x + 1\right )}^{\frac {2}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {1}{{\left (x^2-1\right )}^{2/3}\,{\left (x+1\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {2}{3}} \left (x + 1\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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